Computing the Differential Characteristics of Isointensity Surfaces

In this paper, we present a new method to compute the differential characteristics of isointensity surfaces from three-dimensional images. We show applications where those differentials properties are used to extract characteristic lines from 3D images, called crest lines. The crest lines extracted from different images of the same object are then registered to demonstrate the precision and robustness of our computation. Those experiments also show the direct correspondence between geometrical and anatomical features, for medical images. To compute the differential characteristics of surfaces, such as the principal curvatures, and directions, a traditional approach is to fit a parametric surface model to the 3D image and then to compute the differential characteristics of the surface in the local coordinate system. On the contrary, our method is based on the implicit representation of the surface, and the differential values of the isointensity surfaces are directly computed from the voxel image, without extracting any surface first. In our method, the principal curvatures and directions equations have been derived from the implicit functions theorem, leading to entirely new formulas, which make use of only the differentials of the 3D image, and which allow us to get rid of the problem of parametrizing the surfaces.